Here we sketch the relation between the Holder exponent H of a function and the box-counting dimension of the graph of the function. |
To simplify the calculation, suppose the function f(x) is defined for
|
Here are the basic steps of the argument. |
(1) Divide |
(2) Above each of these intervals, mark off a column of width r. |
(3) In this situation the condition dY = (dt)H means in each of these columns, the graph of f(x) passes through a height of about rH. |
(4) So the number of boxes needed to cover the part of the graph in that column is
about |
(5) The number of these columns is about 1/r. |
(6) The total number of boxes of side r needed to cover the graph is rH-1⋅(1/r) = rH-2. |
(7) Then the box-counting dimension of the graph is about
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Return to Dimensions of UniFractal Cartoons.